Being a aggressive mob, it's going to be hard to kill it as it is about the same speed as you and it can destroy buildings.
When farming for fur, make sure that you have atleast 2 inventory slots open or one of your slots have raw meat or one of them already have mammoth fur. Otherwise, the fur will disappear as the raw meat is prioritized over the fur.
It's recommended to kill a mammoth by leading it into the forest and killing it. Since mammoths can destroy walls, you can still use this method: Put something between you two such as a rock, tree or really anything and kill it with a spear. You can't stay between the border of the forest and winter because the mammoth will follow you into the forest biome but you have a chance since you can hold right click on your mouse on any biome that doesn't slow you down and easily kill it, also the mammoth doesn't have a large hitbox.
Please note that after leading mammoth to the forest, it can run away back to the winter if no players will be close enough.